Online Social Networks and Media

Transcription

1 Online Social Networks and Media Absorbing Random Walks Link Prediction

2 Why does the Power Method work? If a matrix R is real and symmetric, it has real eigenvalues and eigenvectors: λ, w, λ 2, w 2,, (λ r, w r ) r is the rank of the matrix λ λ 2 λ r The vector space of R is the set of vectors that can be written as a linear combination of its rows (or columns) The eigenvectors w, w 2,, w r of R define a basis of the vector space For any vector x, Rx = α w + a 2 w a r w r After t multiplications we have: R t x = λ t α w + λ 2 t a 2 w λ 2 t a r w r Normalizing leaves only the term w.

3 ABSORBING RANDOM WALKS LABEL PROPAGATION OPINION FORMATION ON SOCIAL NETWORKS

4 Random walk with absorbing nodes What happens if we do a random walk on this graph? What is the stationary distribution? All the probability mass on the red sink node: The red node is an absorbing node

5 Random walk with absorbing nodes What happens if we do a random walk on this graph? What is the stationary distribution? There are two absorbing nodes: the red and the blue. The probability mass will be divided between the two

6 Absorption probability If there are more than one absorbing nodes in the graph a random walk that starts from a non-absorbing node will be absorbed in one of them with some probability The probability of absorption gives an estimate of how close the node is to red or blue

7 Absorption probability Computing the probability of being absorbed: The absorbing nodes have probability of being absorbed in themselves and zero of being absorbed in another node. For the non-absorbing nodes, take the (weighted) average of the absorption probabilities of your neighbors if one of the neighbors is the absorbing node, it has probability Repeat until convergence (= very small change in probs) P Red Pink = 2 3 P Red Yellow + 3 P(Red Green) 2 P Red Green = 4 P Red Yellow + 4 P Red Yellow =

8 Absorption probability Computing the probability of being absorbed: The absorbing nodes have probability of being absorbed in themselves and zero of being absorbed in another node. For the non-absorbing nodes, take the (weighted) average of the absorption probabilities of your neighbors if one of the neighbors is the absorbing node, it has probability Repeat until convergence (= very small change in probs) P Blue Pink = 2 3 P Blue Yellow + 3 P(Blue Green) 2 P Blue Green = 4 P Blue Yellow + 2 P Blue Yellow = 3 2 2

9 Why do we care? Why do we care to compute the absorbtion probability to sink nodes? Given a graph (directed or undirected) we can choose to make some nodes absorbing. Simply direct all edges incident on the chosen nodes towards them. The absorbing random walk provides a measure of proximity of non-absorbing nodes to the chosen nodes. Useful for understanding proximity in graphs Useful for propagation in the graph E.g, on a social network some nodes have high income, some have low income, to which income class is a non-absorbing node closer?

10 Example In this undirected graph we want to learn the proximity of nodes to the red and blue nodes 2 2 2

11 Example Make the nodes absorbing 2 2 2

12 Absorption probability Compute the absorbtion probabilities for red and blue P Red Pink = 2 3 P Red Yellow + 3 P(Red Green) P Red Green = 5 P Red Yellow + 5 P Red Pink + 5 P Red Yellow = 6 P Red Green + 3 P Red Pink P Blue Pink = P Red Pink P Blue Green = P Red Green 2 2 P Blue Yellow = P Red Yellow

13 Penalizing long paths The orange node has the same probability of reaching red and blue as the yellow one P Red Orange = P Red Yellow P Blue Orange = P Blue Yellow Intuitively though it is further away

14 Penalizing long paths Add an universal absorbing node to which each node gets absorbed with probability α. With probability α the random walk dies α With probability (-α) the random walk continues as before The longer the path from a node to an absorbing node the more likely the random walk dies along the way, the lower the absorbtion probability α -α α -α -α -α α P Red Green = ( α) 5 P Red Yellow + 5 P Red Pink + 5

15 Propagating values Assume that Red has a positive value and Blue a negative value Positive/Negative class, Positive/Negative opinion We can compute a value for all the other nodes in the same way This is the expected value for the node V(Pink) = 2 3 V(Yellow) + 3 V(Green) V Green = 5 V(Yellow) + 5 V(Pink) V Yellow = 6 V Green + 3 V(Pink)

16 Electrical networks and random walks Our graph corresponds to an electrical network There is a positive voltage of + at the Red node, and a negative voltage - at the Blue node There are resistances on the edges inversely proportional to the weights (or conductance proportional to the weights) The computed values are the voltages at the nodes V(Pink) = 2 3 V(Yellow) + 3 V(Green) V Green = 5 V(Yellow) + 5 V(Pink) V Yellow = 6 V Green + 3 V(Pink)

17 Opinion formation The value propagation can be used as a model of opinion formation. Model: Opinions are values in [-,] Every user u has an internal opinion s u, and expressed opinion z u. The expressed opinion minimizes the personal cost of user u: c z u = s u z u 2 + v:v is a friend of u w u z u z v 2 Minimize deviation from your beliefs and conflicts with the society If every user tries independently (selfishly) to minimize their personal cost then the best thing to do is to set z u to the average of all opinions: z u = s u + v:v is a friend of u w u z u + v:v is a friend of u w u This is the same as the value propagation we described before!

18 Example Social network with internal opinions 2 s = +0.5 s = s = -0.3 s = +0.2 s = -0.

19 Example One absorbing node per user with value the internal opinion of the user s = +0.5 One non-absorbing node per user that links to the corresponding absorbing node s = +0.8 The external opinion for each node is computed using the value propagation we described before Repeated averaging z = z = z = z = 0.04 z = -0.0 s = -0.3 Intuitive model: my opinion is a combination of what I believe and what my social network believes. s = -0.5 s = -0.

20 Transductive learning If we have a graph of relationships and some labels on some nodes we can propagate them to the remaining nodes Make the labeled nodes to be absorbing and compute the probability for the rest of the graph E.g., a social network where some people are tagged as spammers E.g., the movie-actor graph where some movies are tagged as action or comedy. This is a form of semi-supervised learning We make use of the unlabeled data, and the relationships It is also called transductive learning because it does not produce a model, but just labels the unlabeled data that is at hand. Contrast to inductive learning that learns a model and can label any new example

21 Implementation details Implementation is in many ways similar to the PageRank implementation For an edge (u, v)instead of updating the value of v we update the value of u. The value of a node is the average of its neighbors We need to check for the case that a node u is absorbing, in which case the value of the node is not updated. Repeat the updates until the change in values is very small.

22 LINK PREDICTION

23 The Problem Link prediction problem: Given the links in a social network at time t, predict which edges that will be added to the network Which features to use? User characteristics (profile), network interactions, topology Different from the problem of inferring missing (hidden) links (there is a temporal aspect, uses a static snapshot) To save experimental effort in the laboratory or in the field

24 Applications Recommending new friends on online social networks. Predicting the participants or actors in events Suggesting interactions between the members of a company/organization Predicting connections between members of terrorist organizations who have not been directly observed to work together Suggesting collaborations between researchers based on co-authorship. Network evolution model

25 Link Prediction Unsupervised (usually, assign scores based on similarity of endpoints) Supervised (given some positive (created edges) and negative examples (nonexistent edges) Classification Problem Problem: Class imbalance Instead of 0/, rank each edge by its probability to appear in the network

26 D. Liben-Nowell, D. and J. Kleinberg, The link-prediction problem for social networks. Journal of the American Society for Information Science and Technology, 58(7) (2007)

27 The Problem Link prediction problem: Given the links in a social network at time t, predict the edges that will be added to the network during the time interval from time t to a given future time t Which features to use? Based solely on the topology of the network (social proximity) (the more general problem also considers attributes of the nodes and links)

28 Problem Formulation I Consider a social network G = (V, E) where each edge e = <y, v> E represents an interaction between u and v that took place at a particular time t(e) (multiple interactions between two nodes as parallel edges with different timestamps) G[t, t ]: subgraph of G consisting of all edges with a timestamp between t and t, t < t, For four times, t 0 < t 0 < t < t, Given G[t 0, t 0 ], we wish to output a list of edges not in G[t 0, t 0 ] that are predicted to appear in G[t, t ] [t 0, t 0 ] training interval [t, t ] test interval

29 Problem Formulation II What about new nodes (node not in the training interval)? Two parameters: κ training and κ test Core: all nodes that are incident to at least κ training edges in G[t 0, t 0 ], and at least κ test edges in G[t, t ] Predict new edges between the nodes in Core

30 Example Dataset: co-authorship t 0 = 994, t 0 = 996: training interval -> [994, 996] t = 997, t = 999: test interval -> [997, 999] - G collab = <A, E old > = G[994, 996] - E new : authors in A that co-author a paper during the test interval but not during the training interval κ training = 3, κ test = 3: Core consists of all authors who have written at least 3 papers during the training period and at least 3 papers during the test period Predict E new

31 How to Evaluate the Prediction Each link predictor p outputs a ranked list L p of pairs in A A E old : predicted new collaborations in decreasing order of confidence Actual edges: E new = E new (Core Core), n = E new Evaluation method: Size of the intersection of the first n edge predictions from L p that are in Core Core (predicted) and the set E new (actual) How many of the top-n predictions are correct (precision?)

32 Methods for Link Prediction Assign a connection weight score(x, y) to each pair of nodes <x, y> based on the input graph (G collab ) and produce a ranked list of decreasing order of score How to assign the score between two nodes x and y? Some form of similarity or node proximity Most measures focus on the giant component

33 Methods for Link Prediction: Shortest Path For x, y A A E old, score(x, y) = (negated) length of shortest path between x and y If there are more than n pairs of nodes tied for the shortest path length, order them at random. Geodesic distance: number of edges in the shortest path

34 Methods for Link Prediction: Neighborhood-based The larger the overlap of the neighbors of two nodes, the more likely to be linked in the future Let Γ(x) denote the set of neighbors of x in G collab Common neighbors: A adjacency matrix -> A x,y 2 Number of different paths of length 2 Jaccard coefficient: The probability that both x and y have a feature f, for a randomly selected feature that either x or y has

35 Methods for Link Prediction: Neighborhood-based Adamic/Adar: Assigns large weights to common neighbors z of x and y which themselves have few neighbors (weight rare features more heavily) connections to unpopular nodes are more relevant

36 Methods for Link Prediction: Neighborhood-based Preferential attachment: the probability that a new edge has node x as its endpoint is proportional to Γ(x), i.e., nodes like to form ties with popular nodes Researchers found empirical evidence to suggest that co-authorship is correlated with the product of the neighborhood sizes

37 Methods for Link Prediction: based on the ensemble of all paths Not just the shortest, but all paths between two nodes

38 Methods for Link Prediction: based on the ensemble of all paths Katz β measure: Sum over all paths of length l, β > 0 is a parameter of the predictor, exponentially damped to count short paths more heavily Small β predictions much like common neighbors. Unweighted version, in which path () x,y =, if x and y have collaborated, 0 otherwise 2. Weighted version, in which path () x,y = #times x and y have collaborated

39 Methods for Link Prediction: based on the ensemble of all paths Consider a random walk on G collab that starts at x and iteratively moves to a neighbor of x chosen uniformly at random from Γ(x). The Hitting Time H x,y from x to y is the expected number of steps it takes for the random walk starting at x to reach y. score(x, y) = H x,y (symmetric version) The Commute Time C x,y from x to y is the expected number of steps to travel from x to y and from y to x score(x, y) = (H x,y + H y,x ) Can also consider stationary-normed versions: score(x, y) = H x,y π y score(x, y) = (H x,y π y + H y,x π x )

40 Methods for Link Prediction: based on the ensemble of all paths The hitting time and commute time measures are sensitive to parts of the graph far away from x and y -> periodically reset the walk Random walk on G collab that starts at x and has a probability of α of returning to x at each step. Rooted (Personalized) Page Rank: Starts from x, with probability ( a) moves to a random neighbor and with probability a returns to x score(x, y) = stationary probability of y in a rooted PageRank

41 Methods for Link Prediction: based on the ensemble of all paths SimRank score(x, y) = similarity(x, y) The expected value of γ l where l is a random variable giving the time at which random walks started from x and y first meet

42 Methods for Link Prediction: High-level approaches Low rank approximations A adjacency matrix Apply SVD (singular value decomposition) The rank-k matrix that best approximates A

43 Methods for Link Prediction: High-level approaches Unseen Bigrams Unseen bigrams: pairs of word that co-occur in a test corpus, but not in the corresponding training corpus Not just score(x, y) but score(z, y) for nodes z that are similar to x S x (δ) the δ nodes most related to x

44 Methods for Link Prediction: High-level approaches Clustering Compute score(x, y) for al edges in E old Delete the (-p) fraction of these edges for which the score is the lowest, for some parameter p Recompute score(x, y) for all pairs in the subgraph

45 Evaluation: baseline Baseline: random predictor Randomly select pairs of authors who did not collaborate in the training interval Probability that a random prediction is correct, Number of possible predictions Correct predictions In the datasets, from 0.5% (cond-mat) to 0.48% (astro-ph)

46 Evaluation: Factor improvement over random

47 Evaluation: Factor improvement over random

48 Evaluation: Average relevance performance (random) average ratio over the five datasets of the given predictor's performance versus a baseline predictor's performance. The error bars indicate the minimum and maximum of this ratio over the five datasets. The parameters for the starred predictors are as follows: () for weighted Katz, β= 0.005; (2) for Katz clustering, β = 0.00; ρ = 0.5; β2 = 0.; (3) for low-rank inner product, rank = 256; (4) for rooted Pagerank, α = 0.5; (5) for unseen bigrams, unweighted common neighbors with δ = 8; and (6) for SimRank, γ = 0.8.

49 Evaluation: Average relevance performance (distance)

50 Evaluation: Average relevance performance (neighbors)

51 Evaluation: prediction overlap How much similar are the predictions made by the different methods (common predictions)? Why? correct

52 Evaluation: datasets How much does the performance of the different methods depends on the dataset? (rank) On 4 of the 5 datasets best at an intermediate rank On qr-qc, best at rank, does it have a simpler structure? On hep-ph, preferential attachment the best Why is astro-ph difficult? The culture of physicists and physics collaboration

53 Evaluation: small world The shortest path even in unrelated disciplines is often very short

54 Evaluation: restricting to distance three Many pairs of authors separated by a graph distance of 2 who will not collaborate and many pairs who collaborate at distance greater than 2 Disregard all distance 2 pairs

55 Evaluation: the breadth of data Three additional datasets. Proceedings of STOC and FOCS 2. Papers for Citeseer 3. All five of the arxiv sections Common neighbors vs Random

56 Future Directions Improve performance. Even the best (Katz clustering on gr-qc) correct on only about 6% of its prediction Improve efficiency on very large networks (approximation of distances) Treat more recent collaborations as more important Additional information (paper titles, author institutions, etc) To some extent latently present in the graph

57 Future Directions Consider bipartite graph (e.g., some form of an affiliation network) author paper Coauthor graph Apply classification techniques from machine learning A simple binary classification problem: given two nodes x and y predict whether <x, y> is or 0

58

59 Aaron Clauset, Cristopher Moore & M. E. J. Newman. Hierarchical structure and the prediction of missing links in network, Nature, 453, 98-0 (2008)

60 Hierarchical Random Graphs Graph G with n nodes Dendrogram D a binary tree with n leaves Each internal node corresponds to the group of nodes that descend from it Each internal node r of the dendrogram is associated with a probability p r that a pair of vertices in the left and right subtrees of that node are connected Given two nodes i and j of G the probability p ij that they are connected by an edge is equal to p r where r is their lowest common ancestor

61 Hierarchical Random Graphs Example graph Possible dendrogram Assortativity (dense connections within groups of nodes and sparse between them) -> probabilities p r decrease as we move up the tree Given D and the probabilities p r, we can generate a graph, called a hierarchical random graph D: topological structure and parameters {p r }

62 Hierarchical Random Graphs Use to predict missing interactions in the network Given an observed but incomplete network, generate a set of hierarchical random graphs (i.e., a dendrogam and the associated probabilities) that fit the network (using statistical inference) Then look for pair of nodes that have a high probability of connection Is this better than link prediction? Experiments show that link prediction works well for strongly assortative networks (e.g, collaboration, citation) but not for networks that exhibit more general structure (e.g., food webs)

63 A rough idea of how to generate the model r a node in dendrogram D E r the number of edges in G whose endpoints have r as their lowest common ancestor in D, L r and R r the numbers of leaves in the left and right subtrees rooted at r Then the likelihood of the hierarchical random graph is If we fix the dendrogram D, it is easy to find the probabilities {p r } that maximize L(D, {p r }). For each r, they are given by the fraction of potential edges between the two subtrees of r that actually appear in the graph G. L(D) = (/9)(8/9) 8 = L(D2) = (/3)(2/3) 2 (/4) 2 (3/4) 6 =

64 A rough idea of how to generate the model Sample dendrograms D with probability proportional to their likelihood Choose an internal node uniformly at random and consider one of the two ways to reshuffle Always accept the transition if it increases the likelihood else accept with some probability

65 How to Evaluate the Prediction (other) An undirected network G(V, E) Predict Missing links (links not in E) To test, randomly divide E into a training set E T and a probe (test) set E P Apply standard techniques (k-fold cross validation) Each time we randomly pick a missing link and a nonexistent link to compare their scores If among n independent comparisons, there are n times the missing link having a higher score and n times they have the same score, the AUC value is the probability that a randomly chosen missing link is given a higher score than a randomly chosen nonexistent link If all the scores are generated from an independent and identical distribution, the AUC value should be about 0.5.

66 How to Evaluate the Prediction (other) Algorithm assigns scores of all non-observed links as s2 = 0.4, s3 = 0.5, s4 = 0.6, s34 = 0.5 and s45 = 0.6. To calculate AUC, compare the scores of a probe (missing) link and a nonexistent link. (n=) 6 pairs: s3 > s2, s3 < s4, s3 = s34, s45 > s2, s45 = s4, s45 > s34. AUC = ( )/